Methods of simulating wound healing and associated inflammation using agent-based models and, optionally mathematical (differential equation-based) modeling are described herein.
Various approaches have been used to construct simulations of complex biologic processes. All these methods have distinct advantages and disadvantages. In vitro biological systems work well in many situations, but require physical facilities and often are not complex enough or accurate enough to effectively model in vivo systems. In silico (computer simulated) systems are becoming more sophisticated and, as described below, are becoming increasingly able to model biological systems. Two modeling systems are commonly, but not exclusively, utilized to model biological systems: equation-based modeling (EBM), or more specifically ordinary differential equations (ODE), and agent-based modeling (ABM).
The ODE type of modeling is a type of equation-based modeling that consists of establishing a series of differential equations that describe the sequential change in the states of the components of the system over time. The differential equations are derived from known and hypothesized kinetics of the components of the biologic system. This approach has been used for many years to describe chemical systems, for example Michaelis-Menten kinetics. The variables of the equations generally represent average concentrations of the various components. These systems of equations are generally most accurate in settings in which large numbers of individuals of these components are assumed to exist and to exert their effects in aggregate. When the numbers become small, differential equation descriptions break down. The behavior of the system with limited spatial information (e.g. compartments) can be characterized with ODE; if more precise spatial resolution is desired, partial differential equations (PDE) are more commonly used. If simple enough, ODE can be solved analytically. If not, they can be easily solved computationally using a variety of commercially available and free software, as well as proprietary designed for specific implementations of ODE models. Additionally, methods from nonlinear analysis can explore the properties of ODE without completely solving them. Because these equations are based on and describe biologic interactions, these models can potentially predict outcomes beyond the range of data on which these models were initially calibrated. In this latter aspect, EBM are different from statistical models. Furthermore, manipulation of a biologic mechanism can be entered into the model and an outcome derived (predicted).
The ABM type of modeling focuses on the rules and mechanisms of behavior of the individual components of a system, and may be more accurate than EBM in settings in which the stochastic actions of these agents is a better approximation of biological reality as compared to the actions of these components in aggregate. The components of a system are classified into types of “agents” by virtue of shared mechanisms that have been identified experimentally. The mechanisms are expressed as a series of conditional (“if-then”) statements, and computer programs are written to describe the rules of behavior. An example would be the sequence of receptor activation involved in neutrophil adhesion. The model defines a “virtual world” based on characteristics of the reference system and generates populations of the various types of agents. The agents interact based on responses (defined by their rule systems) to inputs and outputs from their environment. For example, simulated cells would respond to variables in their immediate neighborhood, representing the extent of a cell's interaction with its extracellular milieu. The agents run in a parallel fashion to simulate simultaneous behavior, and the dynamics of the system are allowed to emerge from the multiple interactions among the agents over time. Consequently, all measured parameters and outcomes from the model are generated by the actions of the agents. The rules governing the behavior of agents should ideally be well-vetted, simple rules. Because ABMs are mechanistic models, any intervention that deals with a defined mechanism in the model can be simulated. Because they are based on rules, ABMs are often more intuitive to non-mathematicians than EBM (Ermentrout, G. B., et al. (1993). Journal of Theoretical Biology, 160(1), 97-133; An G. Agent-based computer simulation and SIRS: building a bridge between basic science and clinical trials. Shock 2001; 16(4):266-73; Vodovotz Y, Clermont G, Chow C, An G. Mathematical models of the acute inflammatory response. Curr Opin Crit. Care 2004; 10:383-90).
Wound healing is a complex, multi-step process that occurs in many tissues and organs in the body. In epithelial tissues, wound healing typically occurs in the following stages: platelet activation and cytokine release, inflammation, re-epithelialization, formation of granulation tissue and angiogenesis, matrix production, and scar formation and a remodeling phase (See, e.g., Hart J. Inflammation. 1: Its role in the healing of acute wounds. J Wound Care 2002; 11(6):205-9; Hart J. Inflammation. 2: Its role in the healing of chronic wounds. J Wound Care 2002; 11(7):245-9; Goldring S R. Inflammatory mediators as essential elements in bone remodeling. Calcif Tissue Int 2003; 73(2):97-100; Guilak F, Fermor B, Keefe F J, Kraus V B, Olson S A, Pisetsky D S, Setton L A, Weinberg J B. The role of biomechanics and inflammation in cartilage injury and repair. Clin Orthop 2004(423): 17-26; Ramadori G, Saile B. Inflammation, damage repair, immune cells, and liver fibrosis: specific or nonspecific, this is the question. Gastroenterology 2004; 127(3):997-1000; Redd M J, Cooper L, Wood W, Stramer B, Martin P. Wound healing and inflammation: embryos reveal the way to perfect repair. Philos Trans R Soc Lond B Biol Sci 2004; 359(1445):777-84 and Diegelmann R F, Evans M C. Wound healing: an overview of acute, fibrotic and delayed healing. Front Biosci 2004; 9:283-9). These processes represent “snapshots” of a continuum, which may take different amounts of time depending on the tissue being examined. Importantly, these ordered steps can become disrupted in many disease settings that are broadly characterized as exhibiting impaired or aberrant wound healing. Typically, in these settings inflammation is also deranged, as might be expected given the linkage between inflammation and wound healing described above.